What Is the Second Law of Black Hole Thermodynamics?

The phrase second law of black hole thermodynamics commonly refers to either the classical area theorem or the generalized second law. The distinction matters: Hawking radiation can reduce a black hole’s horizon area, but it does not violate the generalized law because the entropy carried outside the hole must also be counted.

The classical area theorem

In classical general relativity, the area of an event horizon cannot decrease over time, provided certain conditions hold. These include the null energy condition for matter and assumptions about the spacetime’s causal structure and the behavior of the horizon. The theorem is often associated with Stephen Hawking.

If a black hole absorbs matter or merges with another black hole, its event-horizon area therefore stays the same or increases. The area theorem is a statement about horizon geometry, not a claim that every quantity associated with a black hole must increase.

For a nonrotating, uncharged Schwarzschild black hole of mass $M$, the horizon area is

$$A=\frac{16\pi G^2M^2}{c^4}.$$

Thus, for this type of black hole, increasing the mass increases the area. More generally, the area is determined by the hole’s mass, angular momentum, and charge.

From area to entropy

Black holes also have an entropy proportional to their horizon area. The Bekenstein–Hawking formula is

$$S_{\text{BH}}=\frac{k_{\text{B}}c^3A}{4G\hbar},$$

where $k_{\text{B}}$ is Boltzmann’s constant, $G$ is Newton’s gravitational constant, $c$ is the speed of light, and $\hbar$ is the reduced Planck constant. Consequently, the classical area theorem implies that black-hole entropy does not decrease in the classical setting.

This entropy is not the ordinary entropy of material located on the horizon. It is a thermodynamic quantity associated with the black hole. Its existence is part of why black-hole mechanics is called thermodynamics: black holes have quantities analogous to entropy and temperature, and their changes obey relations resembling thermodynamic laws.

The generalized second law

Quantum field theory in curved spacetime predicts that black holes emit Hawking radiation. As a black hole radiates, it loses mass and its horizon area can shrink. The classical area theorem does not apply unchanged to this semiclassical process.

The appropriate statement is the generalized second law (GSL):

$$S_{\text{gen}}=S_{\text{BH}}+S_{\text{outside}}$$

never decreases, where $S_{\text{outside}}$ is the appropriately defined, renormalized entropy of quantum fields and matter outside the horizon. When the hole emits radiation, its own entropy falls, but the emitted radiation contributes outside entropy. The GSL says that the combined change is nonnegative:

$$\Delta S_{\text{BH}}+\Delta S_{\text{outside}}\geq 0.$$

This is the black-hole version of the ordinary second law, which says that the total entropy of an isolated system does not decrease. The black hole’s entropy alone need not rise once quantum effects are included; the combined entropy is what matters.

Worked example: two equal black holes merge

Consider two widely separated, nonrotating black holes, each of mass $M$. Each has area

$$A=\frac{16\pi G^2M^2}{c^4},$$

so their total initial horizon area is $2A$. If they merge into one black hole, the classical area theorem requires the final horizon area to be at least $2A$ (assuming the theorem’s conditions hold). It cannot be smaller than the sum of the initial areas.

For an illustrative limiting comparison, suppose the final hole is also nonrotating and has mass $M_f$. Its area is $16\pi G^2M_f^2/c^4$. Requiring this area to be at least $2A$ gives

$$\frac{16\pi G^2M_f^2}{c^4}\geq 2\frac{16\pi G^2M^2}{c^4}, \qquad M_f\geq \sqrt{2}\,M.$$

For holes initially at rest and widely separated, the system’s total mass is approximately $2M$, with a correction from gravitational binding energy. In this simplified comparison, the area bound means the final mass cannot be less than $\sqrt{2}M$. The actual final mass and spin depend on the merger dynamics; this example illustrates the area constraint, not a prediction of the detailed outcome.

Common mistakes

• Confusing the area theorem with the GSL. The area theorem is classical and concerns horizon area. The GSL includes outside entropy and applies to the semiclassical description.
• Assuming Hawking radiation contradicts the second law. Radiation can decrease $S_{\text{BH}}$, but the relevant comparison includes the radiation’s entropy outside the hole.
• Treating the theorem as assumption-free. The classical result depends on conditions such as an appropriate energy condition and global assumptions about the spacetime. Quantum effects can violate classical energy conditions, which is one reason the area statement is not the general quantum law.
• Reading area as volume. The entropy is proportional to horizon area, not to the volume inside the black hole.

In short, classical black-hole processes obey a nondecreasing-area rule under suitable assumptions. When quantum effects such as Hawking radiation are included, the more general rule is that black-hole entropy plus outside entropy does not decrease.